Smoothing Directional Vector Fields Using Dual Norms

This paper provides a new variational paradigm to measure the smoothness of unit vector fields on spatial domains, leading to new methods for smoothing and interpolating such datasets. Our point of view is to consider unit vector fields as linear forms acting on reproducing kernel Hilbert spaces of vector fields or tensors and work with the dual norm, leading to new variational problems and algorithms. We prove, in particular, that these variational problems are well-posed, in the sense that optimal solutions exist in the space of unit vector fields. Experimental results are based on synthetic data and diffusion tensor–magnetic resonance imaging datasets.